Answer:
To solve this problem, we'll start by finding the radius of the unit circle and then determine the scale factor needed to achieve the area of the new circle.
1. Find the radius of the unit circle:
The unit circle has a radius of 1.
2. Area of the unit circle:
The area of a circle is given by \( \pi r^2 \).
For the unit circle with \( r = 1 \):
\[
\text{Area} = \pi \cdot 1^2 = \pi
\]
3. Find the radius of the new circle:
We know the area of the new circle is \( 729\pi \) cm².
Using the area formula \( \pi R^2 = 729\pi \), we can solve for the new radius \( R \):
\[
R^2 = 729
\]
\[
R = \sqrt{729} = 27
\]
4. Determine the scale factor:
The scale factor \( k \) is the ratio of the new radius to the original radius.
\[
k = \frac{R}{r} = \frac{27}{1} = 27
\]
Therefore, the scale factor required to dilate the unit circle to a circle with an area of \( 729\pi \) cm² is 27.